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Saltpeter is potassium nitrate, which does not directly cure meats. Bacteria convert nitrate into nitrite, which is the real preservative. Saltpeter can be replaced by a smaller amount of nitrite to get the same curing effect (most commercial cured meats do this), though a prolonged cure that converts nitrate into nitrite can develop more flavor.
Tender ...

I decided to do some more of my own research on this with the nitrate/nitrite confusion. Thanks to the other answerers, that definitely helped give me a good starting point. I'm writing my own answer so I can include some links. I made it a community wiki (seemed like it might be good for this one).
Firstly, from everything I've been able to find online ...

You are about to enter the wonderful world of Charcuterie, the preserving and curing of meats.
The traditional cut of beef to turned into corned beef (or pastrami, which has a similar preparation) is a well-marbled brisket. I'd imagine that a flank steak or other similar long-and-moist cuts would work out too.
The curing process involves soaking the ...

After your brisket is cooked, refrigerate it overnight. This will help it stay solid when you slice it, and will also improve the texture and flavor. Reheat it before serving. If you have time, this will improve many slow cooked foods.

Cauliflower might work. I've put cauliflower in a food processor to get it to a uniform small-ish size, and then stir-fried it until soft, as a low-carb substitute for rice. Maybe you could dice it into small pieces and do the same. Boiling or steaming instead of stir-frying may give you a softer potato-like texture.

I've been making fresh and cured sausage for years. Here are the details on what you are asking. There are 2 types of cure. Commercially, they are now known as Prague Powder #1 and #2. You can find them on any website that sells sausage making supplies (casings, stuffers, etc). #1 is also known as pink curing salt, and is a mixture of 1 oz sodium ...

I think 2. is the most likely. Remember the beef has been brining for ages, a rinse will just remove the salt from the outside, and even then it won't do much.
Personally I wouldn't bother with the sous vide. 'Proper' corned beef should be gently simmered in a covered pot for about 2.5 hours. It doesn't need 'boiling' per se, and certainly not to oblivion!

Starchy is kind of the nature of hash. I am not sure that you would still have a hash if you eliminate the potatoes, but you should get something delicious in any case.
This low carb website suggests using cabbage in a hash-like dish.

I have used diced turnips in dishes as a replacement for potatoes. That may work if you enjoy eating turnips. Of course the taste will be much different than potatoes.
According to Wikipedia (eep, I know... but just for a rough idea):
100 grams of..
Turnips 7g carbs
Potatoes 17g carbs

Take everything I say with a grain of salt (pun intended) as I am writing from Texas where we do nobler things with brisket. A true corned-beef-expert would be of Irish or Jewish descent, and from New York.
The meat should have relatively tough individual fibers that separate easily because of the long, wet cooking. It is pretty difficult to overcook ...

Once it's been brined for a week, I simmer my corned beef in just enough water to cover it, for about 2 1/2 hours, or until it's tender to the fork. I then leave it to cool for at least half an hour before slicing, though it cuts better when cold.

I bought some from my local pharmacy. I just took in my grandmother's recipe, showed it to the pharmacist and he ordered me a bottle; the bottle was really too big for my needs but it keeps really well in the cupboard